We develop criteria for hitting probabilities of anisotropic Gaussian random fields with associated canonical pseudo-metric given by a class of gauge functions. This yields lower and upper bounds in terms of general notions of capacity and Hausdorff measure, respectively, therefore extending the classical estimates with the Bessel–Riesz capacity and the \(\gamma \)-dimensional Hausdorff measure. We apply the criteria to a system of linear stochastic partial differential equations driven by space-time noises that are fractional in time and either white or colored in space.

中文翻译：

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各向异性高斯随机场：命中概率的标准和应用

我们开发了一种由规范函数给定的具有相关规范伪度量的各向异性高斯随机场命中概率的标准。这分别在容量和Hausdorff测度的一般概念上产生了上限和下限，因此用Bessel–Riesz容量和\（\ gamma \）维Hausdorff测度扩展了经典估计。我们将标准应用于由时空为分数，时空为白色或有色的时空噪声驱动的线性随机偏微分方程组。